Inconsistency of ℕ from a not-finitist point of view
Abstract
Considering the set of natural numbers ℕ, then in the context of Peano axioms, starting from inequalities between finite sets, we find a fundamental contradiction, about the existence of ℕ, from a not-finitist point of view. This proof of inconsistency is not-finitist because it involves infinite totalities.But this is natural considering set theory with the axiom of infinity and all elements of a set. On the other hand a finitist proof would imply the end of mathematics as we know it. Anyway, refusing a precise definition of N, then refusing the axiom of infinity, could be a view to avoid this inconsistency. So the axiom of infinity would seem to have a similar role to coherence. It is not demonstrable, but also it cannot be taken as an axiom if one doesn't want a system to be inconsistent. This proof supports finitist approach in a not arbitrary manner.
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International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 8 Issue 10 ǁ December 2023. w w w . i j m r e t . o r g I S S N : 2 4 5 6 - 5 6 2 8 Page 15 Inconsistency of ℕ from a not-finitist point of view Enrico Pier Giorgio Cadeddu (Master of Science, Oristano, Sardinia - Italy) ABSTRACT: Considering the set of natural numbers ℕ , then in the context of Peano axioms, starting from inequalities between finite sets, we find a fundamental contradiction, about the existence of ℕ , from a not-finitist point of view. KEYWORDS -Inconsistency, Peano axioms, Natural numbers set, Not -finitist I.INTRODUCTION A formal system together an interpretation, constituted of an alphabet, grammar, inference rules, axioms, and a reference set, can produce formalized propositions and deductions (theorems) through with a finite number of steps, that is a finitist approach [1, 2]. A system is consistent whether a proposition and its negation are not deduced. Godel's incompleteness theorems [3], developed on the basis of the system of Principia Mathematica including the axiom of infinity, represent a fortress of logic and consistency against inconsistency. But at the same time they represent a prelude of inconsistency. They give us necessary conditions of consistency, not sufficient ones (undecidable propositions and internal not-demonstrable coherence are these necessary conditions). Considering the successor function S(x) and the existence of all natural numbers, in concordance with Peano axioms and the axiom of infinity, we show a contradiction in ℕ , in a not-finitist way, that is thinking to take all natural numbers simultaneously. II.NATURAL NUMBERS SET The existence of ℕ is granted by the axiom of infinity [4, 5, 6]. This existence imply that one of each element of the set, also in an actual sense, so taken all together. A finite set wouldn't admit the Peano axiom: ∀x(S(x)), with S(x) ∈ ℕ , because the greatest number doesn't have a successor into the finite set. All numbers of ℕ are defined by Peano axioms [7, 8, 9], together their proprieties thanks to the axiom of induction. III.A FUNDAMENTAL CONTRADICTION The two sets: {0, {S(x) | x ∈ ℕ }} (with S(x) ∈ ℕ ) and ℕ , are the same set, that is: {0, {S(x) | x ∈ ℕ }} = {x | x ∈ ℕ } = ℕ (1) We know, as it is demonstrable, that: (x ∈ ℕ )(∀x(x < S(x)). That is 0 < 1, 1< 2, …, n < n+1. At the same time we have: {x | x ≤ y} ≠ {x | x ≤ y + 1} ∀y (2) with y+1 = S(y) ∈ ℕ . That is {0, 1, 2, 3} ≠ {0, 1, 2, 3, 4} and so on, for all y. But necessary condition to have all y (that is ∀y) is that at least one of all these sets in (2) exists equal to ℕ , otherwise all y are not taken; the absence of ℕ (all numbers) in (2) would imply that we could add numbers not present in each set in (2) (so, many numbers would be absent in each set). Then, considering all y, then all x, and equation (1), we are considering in (2) a set equal to ℕ . So we have ℕ ≠ ℕ , a contradiction. It is to notice a question: is it necessary to pass through a necessary condition or, directly, do all y imply a set equal to ℕ ? At first sight the answer seems no and yes respectively. CONCLUSION This proof of inconsistency is not-finitist because it involves infinite totalities. But this is natural considering the set theory with the axiom of infinity (all elements of ℕ ). On the other hand a finitist proof would imply the end of mathematics as we know it. Anyway, refusing a precise definition of ℕ , then refusing the axiom of infinity (and Peano
International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 8 Issue 10 ǁ December 2023. w w w . i j m r e t . o r g I S S N : 2 4 5 6 - 5 6 2 8 Page 16 axioms?), could be a view to avoid this inconsistency. So the axiom of infinity would seem to have a similar role to coherence. It is not demonstrable, but also it cannot be taken as an axiom if one doesn't want a system to be inconsistent. This proof supports finitist approach in a not arbitrary manner and all theories implying ℕ with the axiom of infinity could be revisited (including Godel’s theorems). REFERENCES [1] Jacques Herbrand. Sur la théorie de la démonstration. Cambridge, 12, 1971. [2] Richard Zach. Numbers and functions in hilbert’s finitism. 1998. [3] Kurt Gödel. On formally undecidable propositions of principia mathematica and related systems i 1 (1931). In Godel’s Theorem in Focus, pages 17–47. Routledge, 2012. [4] Jerzy Pogonowski. “mathematics is the logic of the infinite”: Zermelo’s project of infinitary logic. Studies in Logic, Grammar and Rhetoric, 66(3):673–708, 2021. [5] Bertrand Russell. Introduction to mathematical philosophy. Taylor & Francis, 2022. [6] Ernst Zermelo. Investigations in the foundations of set theory i. From Frege to Gödel, pages 199–215, 1908. [7] Francesco Ciraulo. Elementi di logica matematica. [8] Yiannis Moschovakis. The natural numbers. Notes on Set Theory, pages 51–70, 2006. [9] Giuseppe Peano. Arithmetices principia: Nova methodo exposita. Fratres Bocca, 1889.